1. The problem gives the function $g_r(r) = \sqrt{r^2 + 1}$ and some values $g_r(0)=3$, $g_r(1)=\sqrt{r}$. However, the values are inconsistent with the function $g_r(r)$ as defined. 2. We'll analyze the function $g_r(r) = \sqrt{r^2 + 1}$ and check $g_r\circ g_r(r)$. 3. First, we find $g_r\circ g_r(r) = g_r(g_r(r))$. Let $x = g_r(r) = \sqrt{r^2 + 1}$. Then, $$g_r\circ g_r(r) = g_r(x) = \sqrt{x^2 + 1} = \sqrt{(\sqrt{r^2 + 1})^2 + 1} = \sqrt{r^2 + 1 + 1} = \sqrt{r^2 + 2}.$$ 4. The problem also gives $g_r\circ g_r(r) = \frac{r^2 + 1}{r^2 + 1} + 1$ for $r \neq -1$. Simplifying this expression, $$\frac{r^2 + 1}{r^2 + 1} + 1 = 1 + 1 = 2.$$ 5. We notice a discrepancy because from the composition we found $g_r\circ g_r(r) = \sqrt{r^2 + 2}$, while the problem states $g_r\circ g_r(r) = 2$ for $r \neq -1$. 6. To reconcile, note that $\sqrt{r^2 + 2} = 2$ if and only if $r^2 + 2 = 4$, i.e., $r^2 = 2$, so $r = \pm \sqrt{2}$. So the equality holds only at these points, not for all $r$. Final conclusion: The statement $g_r\circ g_r(r) = \frac{r^2 + 1}{r^2 + 1} + 1$ is only valid at $r = \pm \sqrt{2}$ and $r \neq -1$. Otherwise, the composition evaluates to $\sqrt{r^2 + 2}$.